A Compact Notation for Peculiar Properties Characterizing Integer Tetration

A COMPACT NOTATION FOR PECULIAR PROPERTIES CHARACTERIZING INTEGER TETRATION Marco Ripà ,Gabriele Di Pietro Abstract. By initially working in the decimal numeral system, we introduce a compact notation to express the congruence speed of an integer tetration base a, along with the cycle of the rightmost non-stable digits of bafor unit increments of b. The resulting discrete function provides a useful tool for efficiently computing the exact number of frozen digits that characterize the right tail of each nontrivial integer tetration. We also establish an improved upper bound for the minimum hyperexponent ¯ b(a)that guarantees the constancy of the congruence speed of afor all heights b≥¯ b(a). Moreover, we prove that the minimum between the constant congruence speeds of any two integers greater than 1, whose product is not divisible by 10, is always less than or equal to the constant congruence speed of their product. Additionally, still assuming radix-10, we give examples of infinitely many perfect powers whose degree matches their constant congruence speed at every height above 2, emphasizing the peculiar recurrence relations of hyper-4. Finally, Appendix 3 generalizes the described radix-10 framework to all squarefree numeral systems, showing that only in such systems the congruence speed stabilizes to a fixed (positive) value for all a > 1not divisible by the radical of the radix. Furthermore, we derive compact formulas for all prime-radix numeral systems and for the composite squarefree senary case. 2020 Mathematics Subject Classification: 11A07, 11N75 (Primary); 11B50, 13A05 (Secondary). 1. Introduction Let a≥2and b≥1be integers, and assume radix-10. Consider the tetration ba:=(aif b= 1 a((b−1)a)if b≥2 (with the convention 0a= 1 and −1a= 0, adopted solely for notational consistency with [7]) which, for each positive integer n, is known to eventually become periodic modulo 10n(see [10]). This result generally holds for sufficiently large values of the hyperexponent b. For example, for a= 27057 and n= 25, we must reach height 6since 127057 ≡(00000000000000000000)27057 (mod 1025), 227057 ≡4777971988283521495227057 (mod 1025), 327057 ≡(0)545271142050361495227057 (mod 1025), 427057 ≡6344863520050361495227057 (mod 1025), 527057 ≡6449963520050361495227057 (mod 1025), and finally we get 627057 (mod 1025) = 727057 (mod 1025) = 1449963520050361495227057. In detail, we observe that the number of “new” rightmost frozen digits of b27057 is equal to 5for b≤4and decreases to 4for all b≥5. In the 2011 book “La strana coda della serie nn...n ”, the concept of congruence speed was introduced to quantify the growth rate of the number of stable digits that appear at the end of bafor each unit increase in b. That work also established a connection between the congruence speed and the periodic behavior of the least significant digit of bathat is not yet stable at height b. Accordingly, we track the cycle of the rightmost non-stable digit of baat successive heights b, b + 1, b + 2, . . . via the phase shift array (see Section 3 of [8]). With this preprint, we propose a new notation system that efficiently describes the congruence speed of nonnegative integer tetration bases at each height, along with its associated phase shift. Building upon previous studies in the field, we refine and extend key results, including bounds for the minimum hyperexponent ¯ b(a)that ensures stability in the congruence speed of a. Our approach relies on modular arithmetic and p-adic analysis to establish rigorous characterizations of congruence speed behaviors, particularly in the context of bases not divisible by 10. To this end, we include Appendix 3 proving that the constant congruence speed of any integer greater than 1and not divisible by 10 is greater than or equal to the lowest constant congruence speed among its factors. Then, we give infinitely many perfect powers whose degree equals their congruence speed at every height greater than 2(in these cases, the congruence speed remains stable for all hyperexponents greater than 1). 1 2 Lastly, Appendix 3 extends the decimal framework to all squarefree numeral systems, providing explicit formulas for each prime case and, in addition, for the senary numeral system. It also formulates the conjecture that squarefreeness is the structural criterion governing when each tetration base coprime to the radix has a constant congruence speed and, in the subsequent Remarks 5 and 6, asserts that only in such numeral systems do all bases greater than 1and not divisible by the radical of the selected numeral system exhibit a (strictly positive) constant congruence speed. Taken together, these results establish a unified framework for the stabilization of rightmost digits in tetration, showing that the constant congruence speed is a characteristic property of all and only squarefree numeral systems. 2. Congruence speed and phase shift of tetration The present section provides rigorous definitions of the terminology introduced earlier, establishing a formal structure for a more compact and efficient notation to describe the recurrence properties characteristic of the right tail of ba. For clarity, we denote N0as the set of nonnegative integers (including zero) and Nas the set of strictly positive integers (i.e., {1,2,3, . . .}). In [5], the congruence speed of each nonnegative integer awas originally defined as the function V:N0×N−→ N0 (a, b)7→ V(a, b), and the author specified that if the constraint 10 ∤ais imposed along with the sufficient condition b>a, the congruence speed no longer depends on b, so we obtain a constant congruence speed, the one entry function V:N\ {multiples of 10}−→N0 a7→ V(a) peculiar of hyper-4and fully described by Equations (3) and (16) of [6]. Since a casual reader could easily be misled by the use of V(a, b)and V(a), and given that describing the congruence speed of aas V(a, 1),V(a, 2),. . .,V(a, a), and V(a, a+1) = V(a)is not very elegant, let us rewrite the original definitions of congruence speed and constant congruence speed as follows. Definition 2.1. Let n∈N0and assume that a∈N\ {1}is not a multiple of 10. Then, given b−1a≡ba (mod 10n)∧b−1a≡ ba(mod 10n+1), for each b= 1,2,3, . . .,vb(a)returns the strictly positive integer such that ba≡b+1a(mod 10n+vb(a))∧ba≡ b+1a(mod 10n+vb(a)+1), and we define vb(a)as the congruence speed of the base aat height b. Definition 2.2. Let a∈N\ {1}not be a multiple of 10. Let ¯ b(a):= min{b∈N:vb(a)=vb+k(a),∀k∈N0}. We define as the constant congruence speed of athe nonnegative integer v¯ b(a):=v¯ b(a)(a). Assuming b > 2, we note that v¯ b(a) = v¯ b+1(a) = v¯ b+2(a) = · · · and vb(a)≥vb+1(a)≥vb+2(a)≥ · · · hold for each agreater than 1and not divisible by 10 (see Equations (17) and (18) of [9]). Now, let us compactly denote ע(a)as the congruence speed of the integer tetration base a. Consequently, if ais greater than 1and is not a multiple of 10, we can write ע(a):= (v1(a), v2(a), . . . , v¯ b−1(a); v¯ b(a)) since we indicate (v1(a), v2(a), . . . , v¯ b−1(a), v¯ b(a), v¯ b(a), v¯ b(a), . . .)as (v1(a), v2(a), . . . , v¯ b−1(a); v¯ b(a)). Then, trivially, ע(0) = (; 0) and ע(1) = (1; 0). On the other hand, we know that the congruence speed of every tetration base divisible by 10 never becomes stable, so we can proceed as follows. Let νp(. . .)be the p-adic valuation of the argument. We have that the congruence speed of ais given by (2.1) ע(a):=         (v1(a), v2(a), . . . , v¯ b−1(a); v¯ b(a)) if a > 1∧10 ∤a (v1(a), v2(a),...; +∞)if a > 1∧10 |a (; v¯ b(a)) if a= 0 (1; v¯ b(a)) if a= 1 , 3 where (see Equation (3) of [6]) (2.2) v¯ b(a) =                                                      0if a∈ {0,1} min {ν2(a−1), ν5(a−1)}if a≡1 (mod 20) ∧a= 1 min {ν2(a+ 1), ν5(a−1)}if a≡11 (mod 20) ν5a2+ 1if a≡2,8 (mod 10) min ν2(a+ 1), ν5a2+ 1 if a≡3,7 (mod 20) min ν2(a−1), ν5a2+ 1 if a≡13,17 (mod 20) ν5(a+ 1) if a≡4 (mod 10) ν2(a−1) if a≡5 (mod 20) ν2(a+ 1) if a≡15 (mod 20) ν5(a−1) if a≡6 (mod 10) min {ν2(a−1), ν5(a+ 1)}if a≡9 (mod 20) min {ν2(a+ 1), ν5(a+ 1)}if a≡19 (mod 20) . Equation (2.2) follows from the nonzero solutions of the fundamental equation y5=yin the commutative ring of 10-adic integers Z10 := lim ←−n Z 10nZ(see (2.3) below). yi=1,2,...,14 =                                                          α′ 1= 1 −2·52n∞=...15487480163574218751 iff i= 1 α′ 2=25n∞=...07839804103263499879186432 iff i= 2 α′ 3=52n∞−25n∞=...52996418333704193 iff i= 3 α′′ 3=−52n∞−25n∞=...476581907922943 iff i= 4 α′ 4=52n∞−1=...7392256259918212890624 iff i= 5 α′ 5=52n∞=...19977392256259918212890625 iff i= 6 α′′ 5=−52n∞=...022607743740081787109375 iff i= 7 α′ 6= 1 −52n∞=...2607743740081787109376 iff i= 8 α′ 7=−52n∞+25n∞=...003581666295807 iff i= 9 α′′ 7=52n∞+25n∞=...59523418092077057 iff i= 10 α′ 8=−25n∞=...160195896736500120813568 iff i= 11 α′ 9= 2 ·52n∞−1=...84512519836425781249 iff i= 12 α′′ 9=−1 = ...999999999999999999999999999999 iff i= 13 α′′ 1=1=...00000000000000000000000000000001 iff i= 14 .(2.3) Now, as we pick each element of the set {α′ 1, α′ 2, α′ 3, α′′ 3, α′ 4, α′ 5, α′′ 5, α′ 6, α′ 7, α′′ 7, α′ 8, α′ 9, α′′ 9, α′′ 1}and multiply it by each element of the same set (including the original element itself), we obtain Table 1. 4 Table 1. Multiplicative recurrences of the solutions of the 10-adic equation y5=y. ·α′′ 1α′ 1α′ 2α′ 3α′′ 3α′ 4α′ 5α′′ 5α′ 6α′ 7α′′ 7α′ 8α′ 9α′′ 9 α′′ 1α′′ 1α′ 1α′ 2α′ 3α′′ 3α′ 4α′ 5α′′ 5α′ 6α′ 7α′′ 7α′ 8α′ 9α′′ 9 α′ 1α′ 1α′′ 1α′ 2α′′ 3α′ 3α′ 4α′′ 5α′ 5α′ 6α′′ 7α′ 7α′ 8α′′ 9α′ 9 α′ 2α′ 2α′ 2α′ 4α′ 6α′ 6α′ 80 0 α′ 2α′ 4α′ 4α′ 6α′ 8α′ 8 α′ 3α′ 3α′′ 3α′ 6α′ 9α′′ 9α′ 2α′ 5α′′ 5α′ 8α′ 1α′′ 1α′ 4α′′ 7α′ 7 α′′ 3α′′ 3α′ 3α′ 6α′′ 9α′ 9α′ 2α′′ 5α′ 5α′ 8α′′ 1α′ 1α′ 4α′ 7α′′ 7 α′ 4α′ 4α′ 4α′ 8α′ 2α′ 2α′ 60 0 α′ 4α′ 8α′ 8α′ 2α′ 6α′ 6 α′ 5α′ 5α′′ 50α′ 5α′′ 50α′ 5α′′ 50α′′ 5α′ 50α′ 5α′′ 5 α′′ 5α′′ 5α′ 50α′′ 5α′ 50α′′ 5α′ 50α′ 5α′′ 50α′′ 5α′ 5 α′ 6α′ 6α′ 6α′ 2α′ 8α′ 8α′ 40 0 α′ 6α′ 2α′ 2α′ 8α′ 4α′ 4 α′ 7α′ 7α′′ 7α′ 4α′ 1α′′ 1α′ 8α′′ 5α′ 5α′ 2α′ 9α′′ 9α′ 6α′′ 3α′ 3 α′′ 7α′′ 7α′ 7α′ 4α′′ 1α′ 1α′ 8α′ 5α′′ 5α′ 2α′′ 9α′ 9α′ 6α′ 3α′′ 3 α′ 8α′ 8α′ 8α′ 6α′ 4α′ 4α′ 20 0 α′ 8α′ 6α′ 6α′ 4α′ 2α′ 2 α′ 9α′ 9α′′ 9α′ 8α′′ 7α′ 7α′ 6α′ 5α′′ 5α′ 4α′′ 3α′ 3α′ 2α′′ 1α′ 1 α′′ 9α′′ 9α′ 9α′ 8α′ 7α′′ 7α′ 6α′′ 5α′ 5α′ 4α′ 3α′′ 3α′ 2α′ 1α′′ 1 Table 2 (also referenced in Appendix 3) synthesizes the behavior of the last digit/two digits of the congruence classes modulo 10/modulo 20 (respectively) enumerated in (2.2). In detail, we note that all the even entries in Table 2 (i.e., 2,4,6, and 8) are assumed modulo 10, while the remaining entries (i.e., 1,11,3,13,7,17,5, 15,9, and 19) are considered modulo 20. Table 2. Transformation through the product of all pairs of congruence classes modulo 10 or 20 considered in (2.2). ·1 11 2, 8 3, 7 13, 17 4 5 15 6 9 19 11 11 2, 8 3, 7 13, 17 4 5 15 6 9 19 11 11 1 2, 8 13, 17 3, 7 4 15 5 6 19 9 2, 8 2, 8 2, 8 4, 6 4, 6 4, 6 2, 8 0 0 2, 8 2, 8 2, 8 3, 7 3, 7 13, 17 4, 6 9, 1 19, 11 2, 8 15 5 2, 8 3, 7 13, 17 13, 17 13, 17 3,7 4, 6 19, 11 9,1 2, 8 5 15 2, 8 13, 17 3, 7 44 4 2, 8 2, 8 2, 8 6 0 0 4 6 6 55 15 0 15 5 0 5 15 0 5 15 15 15 5 0 5 15 0 15 5 0 15 5 66 6 2, 8 2, 8 2, 8 4 0 0 6 4 4 99 19 2, 8 3, 7 13, 17 6 5 15 4 1 11 19 19 9 2, 8 13, 17 3, 7 6 15 5 4 11 1 As a general result, we note that, assuming q, a q∈N\ {1}:a≡ 0 (mod 10), (2.4) v¯ bq·a q≥minv¯ b(q), v¯ ba q always holds (see Appendix 3 for the proof). Thus, for each tetration base aas above, q|aimplies v¯ b(a)≥minnv¯ b(q), v¯ ba qo. In this regard, we note that if a:= 999 ...999 (repunit 9’s), q:= 111 ...111 (repunit 1’s), and k∈Nare such that 10k−1<q<a<10k, then the difference v¯ b(a)−minnv¯ b(q), v¯ ba qois always equal to k−1(since v¯ b(999 ...999) = v1(10k−1)=kwhile v¯ b(9) = v1(101−1)=1=v2(111 ...111) = v¯ b(111 ...111)). In the end, the constant congruence speed of every integer greater than 1and not divisible by 10 is necessarily greater than or equal to the minimum of the constant congruence speeds of its factors. Remark 1. Despite the asymmetrical nature of relation (2.4), which defines only a (weak) link between the constant congruence speed of a given integer (greater than 1and not a multiple of 10) and its factorization, Section 3 of [6] shows that the only prime numbers greater than 5with a unit constant congruence speed are necessarily congruent to 2,3,4,6,8,9,11,12,13,14,16,17,19,21,22, or 23 modulo 52. Furthermore, Theorem 3 of [5] establishes the existence of infinitely many prime numbers characterized by each positive 5 value of constant congruence speed, so the constraint v¯ b(a)=1is neither a sufficient nor a necessary condition for the primality of a. As a result, if we combine the popular primality criterion that every prime greater than 3is congruent to 1 or 5modulo 6with the additional requirement that the constant congruence speed must equal 1, the observed increase in the frequency of primes (within a range of natural numbers) is merely a trivial consequence of the fact that we are also excluding all multiples of 5from the residual list of candidate primes. We do not explore this aspect further, as the aim of the present remark is to outline a brief application tip of the constant congruence speed formula introduced in [5], beyond its use in recreational mathematics contexts – such as proving that the exact number of stable digits of the well-known Graham’s number, g64, is precisely slog3(g64)−1, where slog3(g64)denotes the base-3super-logarithm of Graham’s number itself [8]. As stated in Definition 3.2 of [8], “(. . . ) we call phase shift of aat height bthe congruence class modulo 10 of the difference between the rightmost non-stable digit of ba”. For brevity purposes, similarly to Definition we set s¯ b(a):=s¯ b(a)(a)and then we can compactly indicate the phase shift of each integer tetration base a∈N0as ס(a), where ס(a)is defined as follows: (2.5) ס(a):=                                                (s1(a), s2(a), . . . , s¯ b−1(a); [s¯ b(a), s¯ b+1(a), s¯ b+2(a), s¯ b+3(a)]) if s¯ b(a)=s¯ b+2(a) (s1(a), s2(a), . . . , s¯ b−1(a); [s¯ b(a), s¯ b+1(a)]) if (s¯ b(a)=s¯ b+2(a)∧s¯ b(a)=s¯ b+1(a)) (s1(a), s2(a), . . . , s¯ b−1(a); [s¯ b(a)]) if s¯ b(a) = s¯ b+1(a) (s1(a), s2(a); [s¯ b(a)]) if 10 |a (; [9,1]) if a= 0 (; [0]) if a= 1 . In particular, we call asymptotic phase shift the array [s¯ b(a), s¯ b+1(a), s¯ b+2(a), s¯ b+3(a)] (or [s¯ b(a), s¯ b+1(a)], or [s¯ b(a)]) of ס(a)(see Section 3 of [8]). In (2.5), the statement “10 |a⇒ס(a) = (s1(a), s2(a); [s¯ b(a)])”, arises from the trivial consideration that ¯ b(a)≤3holds for all a∈ {multiples of 10}(see the comments of the sequence A377124 of the OEIS [3,11]). Assuming that a∈N\ {1}is such that 10 ∤a, from [6], we have that ¯ b(a)−2≤˜ν(a), where (2.6) ˜ν(a):=                        ν5(a−1) if a≡11(mod 20) ν5(a2+ 1) if a≡3,7(mod 10) ν5(a+ 1) if a≡9(mod 20) 2if a= 5 1if (a≡5(mod 10) ∧a= 5) 1if a≡2,6,16,18,19(mod 20) 0if a≡1,4,8,12,14(mod 20) . Accordingly, let ˜ b(a):= ˜ν(a) + 2 and ¯ b(a)≤˜ b(a)follows. Remark 2. In (2.6), we distinguish between the case a= 5 (characterized by ¯ b(a) = 4) and the case a: (a≡5 (mod 10) ∧a= 5) (where ¯ b(a)=3holds). Then, we observe that 5is the only integer greater than 1 whose third tetration is congruent to its second tetration modulo 1plus the number of the digits of its second tetration (i.e., the only solution in N\ {1}of 3a≡2a(mod 10⌊log10(2a)⌋+2)is 5– see [8], page 3). Thus, for each acongruent to 5modulo 10, a sufficient but not necessary condition that ensures the perfect match between the congruence speed and the number of new stable digits of bais given by b>2, so we can finally write ע(5) = (1,4,3; 2) (instead of (1,3,4; 2)) and conclude that baexhibits exactly 1 + 4 + 3 + (b−3) ·2stable digits at heights 3,4,5, . . . (since v1(5) + v2(5) + v3(5) = 1 + 4 + 3 = 1 + 3 + 4,¯ b(5) = 4, and v¯ b(5) = 2). Now, if we are not able to determine the exact value of ¯ b(a)for some given integer tetration base aabove 1, we can still write the congruence speed of aas (2.7) ˜ ע(a):=((v1(a), v2(a), . . . , v˜ b−1(a); v¯ b(a)) if 10 ∤a (v1(a), v2(a),...; +∞)if 10 |a. 6 Similarly, we can express the phase shift of every a∈N\ {1}in the form (2.8) ˜ ס(a):=                              (s1(a), s2(a), . . . , s˜ b−1(a); [[s˜ b(a), s˜ b+1(a), s˜ b+2(a), s˜ b+3(a)]]) if s˜ b(a)=s˜ b+2(a) (s1(a), s2(a), . . . , s˜ b−1(a); [[s˜ b(a), s˜ b+1(a)]]) if (s˜ b(a)=s˜ b+2(a)∧s˜ b(a)=s˜ b+1(a)) (s1(a), s2(a), . . . , s˜ b−1(a); [s˜ b(a)]) if s˜ b(a)=s˜ b+1(a) (s1(a), s2(a); [s3(a)]) if 10 |a , where [[s˜ b(a), s˜ b+1(a), s˜ b+2(a), s˜ b+3(a)]] is always one of the four circular permutations of the asymptotic phase shift of a(i.e., [s¯ b(a), s¯ b+1(a), s¯ b+2(a), s¯ b+3(a)] is necessarily equal to [[s˜ b(a), s˜ b+1(a), s˜ b+2(a), s˜ b+3(a)]], or [[s˜ b+1(a), s˜ b+2(a), s˜ b+3(a), s˜ b(a)]], or [[s˜ b+2(a), s˜ b+3(a), s˜ b(a), s˜ b+1(a)]], or [[s˜ b+3(a), s˜ b(a), s˜ b+1(a), s˜ b+2(a)]], and similarly [s¯ b(a), s¯ b+1(a)] = [[s˜ b(a), s˜ b+1(a)]] or [s¯ b(a), s¯ b+1(a)] = [[s˜ b+1(a), s˜ b(a)]]), while it is trivial to point out that s˜ b(a)=s˜ b+1 implies [s˜ b(a)] = [s¯ b(a)] (so choosing to write [[s˜ b(a)]] instead of [s˜ b(a)] or even [s¯ b(a)] would be pointless). In this case, we call modular phase shift the array [[s˜ b(a), s˜ b+1(a), s˜ b+2(a), s˜ b+3(a)]] (or [[s˜ b(a), s˜ b+1(a)]], or [s˜ b(a)]) of ˜ ס(a). Lastly, for each a:=j·10csuch that j, c ∈N\ {1}(which implies 10 |aby construction), the congruence speed of such tetration bases can be compactly rewritten (for each b∈N) as (2.9) vb(j·10c) = (0iff j= 0 c·b−1(j·10c)−b−2(j·10c)otherwise , by [7], and thus, for positive integers cand j, we can finally state that (2.10) ע(a):=                                  (v1(a), v2(a), . . . , v¯ b−1(a); v¯ b(a)) if a > 1∧10 ∤a (c, c ·0(j·10c)−−1(j·10c), c ·1(j·10c)−0(j·10c), c·2(j·10c)−1(j·10c),...; +∞) if a=j·10c(j, c ∈N) (; v¯ b(a)) if a= 0 (1; v¯ b(a)) if a= 1 . Here, for any integer r > 1, we use the notation νr(d)to denote the largest integer q≥0such that rq|d. This notation is purely exponent-counting and does not define a valuation unless ris a prime number. Thus, νr(d)denotes the exponent of the highest power of rdividing d. For each b>2, we have ba≡b+1a(mod 10Pb k=1 vk(a))∧ba≡ b+1a(mod 10(Pb k=1 vk(a))+1). Consequently, if b > ¯ b(a)and 1<a:a≡ 0 (mod 10) are given, ba≡b+1a(mod 10v1(a)+v2(a)+···+v¯ b(a)−1(a)+(b(a)−(¯ b(a)−1))·v¯ b(a))∧ ba≡ b+1a(mod 10v1(a)+v2(a)+···+v¯ b(a)−1(a)+(b(a)−(¯ b(a)−1))·v¯ b(a)+1) follows by construction. Under the constraints above, the integer tetration baexhibits exactly P¯ b(a)−1 k=1 vk(a)+b−¯ b(a)−1·v¯ b(a) stable digits, and then the bound (b−2) ·v¯ b(a)≤v1(a) + v2(a) + · · · +vb(a)≤(b+1)·v¯ b(a)holds for all such pairs (a, b)(e.g., 4002has exactly 0 + 0 + (400 −2) ·1frozen digits). For example, let c,k, and tbe positive integers such that t≥2+ν10(c). Then, the congruence speed of 10t+k+ 10t−ν10(c)+ 1cis (2.11) ע10t+k+ 10t−ν10(c)+ 1c= (2 ·t;t), 7 which does not depend on k, so (2.11) shows the existence of infinitely many c-th perfect powers whose constant congruence speed is also equal to c(for this purpose, it is sufficient to assume t=cand then v¯ b10k+t+ 10t−ν10(c)+ 1c=v210k+c+ 10c−ν10(c)+ 1c=cfollows – see [9]). An improved version of this result is provided in Remark 3. Remark 3. Let c, k, m + 1, t ∈Nbe such that t≥2 + ν10(c)(i.e., tgreater than 1plus the number of trailing 0’s at the end of the radix-10 expansion of c, if any), k≥t, and assume m≡0,1 (mod 3). Then, 3·m·10t+k+ 10t+k+ 10t−ν10(c)+ 1cis always a c-th perfect power (i.e., exactly a c-th perfect power by the digit sum divisible by 3and not by 32argument) characterized by a constant congruence speed of t. In particular, we observe that, for each positive integer c, also v¯ b10c−ν10(c)+ 1c=c, and then 10c−ν10(c)+ 1cis always a perfect power of degree exactly cwith the same constant congruence speed (by Mihăilescu’s theorem [4] – since 10c−ν10(c)is a perfect power different from 23). The preceding example is not peculiar to the decimal numeral system, but is simply the radix-10 instance of a more general identity. Let rad(r)denote the product of the distinct prime factors of r. For every integer r≥3(squarefree, or non-squarefree except when rad(r)|cand r∤c), and every c≥1,t≥2 + νr(c), and z≥0, we have (2.12) v[r] ¯ bz·rt+1 +rt−νr(c)±1c=v[r] ¯ brt−νr(c)±1c=v[r] 2rt−νr(c)±1c=t, where v[r] ¯ bdenotes the constant congruence speed in radix-r(see Appendix 3 for the general framework in squarefree numeral systems). 3. Conclusion The notation we have introduced for the congruence speed and phase shift of tetration may facilitate more sophisticated investigations of these properties in the future. We expect that this specialized nomenclature has not only supported our proof in Appendix 3, which shows how the relationship between the constant congruence speed and the corresponding multiplicative group ensures that v¯ b(a)≥minv¯ b(q), v¯ ba q holds for each pair q, a qof integers greater than 1and not divisible by 10, but may also inspire further research aimed at providing a comprehensive description of the set Mof all pairs of positive integers whose minimum constant congruence speed equals that of their product. Such a classification would allow us to determine in advance whether v¯ bq·a q= minnv¯ b(q), v¯ ba qois satisfied for the given pair q, a q(at present, for example, we can confirm that (624,22943) ∈Monly after directly computing v¯ b(14316432) = 4 and verifying that min{v¯ b(624), v¯ b(22943)}= min{4,5}= 4 also holds). In the previous sections, we have focused our study of congruence speed and phase shift entirely on radix-10; however, Appendix 3 extends the analysis of the constant congruence speed to all squarefree numeral systems, so that proving the general Conjecture 3.1, stated there, will remain a key objective for future research in the field. References [1] Germain, J. (2009). On the Equation ax≡x(mod b).Integers, 9(6), 629–38. [2] Koblitz, N. (1984) p-adic Numbers, p-adic Analysis, and Zeta-Functions (Second Edition), Springer-Verlag, New York. [3] Finch, S. (2021). On-Line Encyclopedia of Integer Sequences. The Mathematical Intelligencer, 43(2), 146–147. [4] Mihăilescu, P. (2004). Primary Cyclotomic Units and a Proof of Catalan’s Conjecture, Journal für die reine und angewandte Mathematik, 27, 167–195. [5] Ripà, M. (2021). The congruence speed formula, Notes on Number Theory and Discrete Mathematics, 27(4), 43–61. [6] Ripà, M., & Onnis, L. (2022). Number of stable digits of any integer tetration, Notes on Number Theory and Discrete Mathematics, 28(3), 441–457. [7] Ripà, M. (2024). Congruence speed of the tetration bases ending with 0. arXiv.org, Available online at: https://arxiv.org/ pdf/2402.07929. [8] Ripà, M. (2025). Graham’s number stable digits: an exact solution. Notes on Number Theory and Discrete Mathematics, 31(3), 607–616. [9] Ripà, M. (2024). On the relation between perfect powers and tetration frozen digits, Journal of AppliedMath, 2(5), Article 1771. [10] Shapiro, D. B. & Shapiro, S. D. (2007). Iterated exponents in number theory, Integers, 7(1), Article A23. [11] Sloane, N. J. A. (2018). The on-line encyclopedia of integer sequences. Notices of the American Mathematical Society, 65(9), 1062–1074. 8 Appendix A Proof of (2.4. ) This appendix is devoted to proving that the constant congruence speed of each integer greater than 1and not a multiple of 10 is greater than or equal to the minimum of the constant congruence speeds of every subset of its factors whose product equals the given tetration base (so that v¯ bq·a q≥minnv¯ b(q), v¯ ba qo holds for all q, a q∈N\{multiples of 10 } ∪ {1}). For this purpose, given m:=a q, we consider (2.2) to show that (2.4) holds for each pair of positive integers (q, m)whose product q·mis greater than 1and not a multiple of 10. Since, in (2.4), qand m(i.e., a q) can independently span the entire set N\{multiples of 10 } ∪ {1}. This result implies the general property that the constant congruence speed of every integer greater than 1(whose last digit is not 0) cannot be lower than the lowest constant congruence speed among its factors (as we can ideally iterate the same process as many times as needed). Consequently, we will proceed case by case to fulfill all the lines of (2.2). Since all the cases mentioned above flow in the 7types of multiplicative classes listed in Table 3, we write the symbol (r)next to some class products to make clear that, in the corresponding case of the present proof, the variables qand mshould be swapped with each other. Table 3. The seven fundamental multiplicative classes enumerated in (2.2). ·1 11 2, 8 3, 7 13, 17 4 1I I; II III III; II III; I II 11 I; II I; IV III III; IV III; II(r) II 2, 8 III III V; VI V; VI V; VI VII 3, 7 III; I III; IV V; VI IV; V; VI V; II(r); VI VII 13, 17 III; I III; II(r) V; VI V; II(r); VI V; I; VI VII 4II II VII VII VII IV 5I II(r) / II(r) I / 15 II IV / IV II / 6I I III(r) III(r) III(r) II(r) 9II; I II; II(r) VII VII; II(r) VII; I IV 19 II II; IV VII IV; VII VII; II IV Table 3. The seven fundamental multiplicative classes enumerated in (2.2), continued. ·5 15 6 9 19 1I II I II; I II 11 II(r) IV I II; II(r) II; IV 2, 8 / / III(r) VII VII 3, 7 II(r) IV III(r) VII; II(r) IV; VII 13, 17 I II III(r) VII; I VII; II 4/ / II(r) IV IV 5I II / I II 15 II IV / II(r) IV 6/ / I II II 9I II(r) II IV; I IV; II 19 II IV II IV; II IV CLASS I PRODUCTS Let q≡m≡1 (mod 20). We consider the number q·m, which is also congruent to 1modulo 20, aiming to evaluate its constant congruence speed (we are interested in comparing v¯ b(q·m) with v¯ b(q) and v¯ b(m)). With reference to (2.2), we can first suppose that min{ν5(q·m−1), ν2(q·m−1)}=ν5(q·m−1), min{ν2(q−1), ν5(q−1)}=ν5(q−1), and min{ν2(m−1), ν5(m−1)}=ν5(m−1). If so, ν5(q·m−1) can be rewritten as ν5(q−1 + (m−1) ·q)and then, by the properties of p-adics, we get ν5(q−1 + (m−1) ·q)≥min{ν5(q−1), ν5((m−1) ·q)}. Using the properties of p-adics on the product, ν5((m−1) ·q)≥ν5(m−1) 9 easily follows (since ν5((m−1) ·q) = ν5(m−1) + ν5(q)). Thus, the 5-adic valuation of (q·m−1) is greater than or equal to the minimum between ν5(q−1) and ν5(m−1). On the other hand, if min{ν2(q−1), ν5(q−1)}=ν2(q−1) and min{ν2(m−1), ν5(m−1)}=ν2(m−1), the proof would remain unchanged, provided that we explicitly state the inequalities ν5(q−1) ≥ν2(q−1) and ν5(m−1) ≥ν2(m−1). Conversely, if min{ν5(q·m−1), ν2(q·m−1)}=ν2(q·m−1), we can assume that min{ν2(q−1), ν5(q−1)}= ν2(q−1) and also that min{ν2(m−1), ν5(m−1)}=ν2(m−1). In the same way, we observe that ν2(q·m−1) can be rewritten as ν2(q−1 + (m−1) ·q)and then, applying the well-known p-adic properties, we find that the 2-adic valuation of (q·m−1) is greater than or equal to the minimum between ν2(q−1) and ν2(m−1). CLASS II PRODUCTS Here we consider the case where q≡1 (mod 20) and m≡11 (mod 20) (see (2.2)), so the product q·m belongs to the congruence class 11 modulo 20. Then, we assume that min{ν2(q·m+ 1), ν5(q·m−1)}=ν2(q·m+ 1),min{ν2(q−1), ν5(q−1)}=ν2(q−1), and min{ν2(m+ 1), ν5(m−1)}=ν2(m+ 1). At this point, we note how ν2(q·m+ 1) can be rewritten as ν2(m+1+m·(q−1)) so that, from the properties of p-adic valuations, we get ν2(m+1+m·(q−1)) ≥min {ν2(m+ 1), ν2((q−1) ·m)}. Hence, from ν2((q−1) ·m) = ν2(q−1) + ν2(m), it follows that ν2((q−1) ·m)≥ν2(q−1). Thus, the 2-adic valuation of (q·m+ 1) is greater than or equal to the minimum between ν2(m+ 1) and ν2(q−1). On the contrary, if we had supposed that min{ν2(q−1), ν5(q−1)}=ν5(q−1) and min{ν2(m+1), ν5(m−1)}= ν5(m−1), the proof of this case would not have changed substantially, provided that we had explicitly stated that ν2(q−1) ≥ν5(q−1) and ν2(m−1) ≥ν5(m−1). CLASS III PRODUCTS Let q≡1 (mod 20) and m≡2,8 (mod 10). We evaluate q·mknowing that it is congruent to 2or 8 modulo 10. With reference to (2.2), we consider ν5((q·m)2+ 1) and assume that min{ν2(q−1), ν5(q−1)}=ν5(q−1). The other value to be taken into account is ν5(m2+ 1), which is associated with the congruence classes 2and 8modulo 10. We note that ν5((q·m)2+ 1) can be rewritten as ν5((q2−1) ·(m2−1) + q2+m2), which is greater than or equal to min ν5((q2−1) ·(m2−1)), ν5(q2+m2). Since ν5((q2−1)·(m2−1)) = ν5((q−1)·(q+1)·(m−1)·(m+1)), by the properties of the p-adic valuations, ν5((q−1) ·(q+1)·(m−1) ·(m+ 1)) ≥ν5(q−1) easily follows. At the same time, from ν5(q2+m2)=ν5(m2+1+q2−1), we have that ν5(m2+1+q2−1) ≥ min ν5(m2+ 1), ν5(q2−1)and, since ν5(q2−1) = ν5((q−1) ·(q+ 1)) = ν5(q−1) + ν5(q+ 1), it follows that ν5(q2−1) ≥ν5(q−1). Thus, the 5-adic valuation of ((q·m)2+ 1) is greater than or equal to the minimum between ν5(q−1) and ν5(m2+ 1). In contrast, if min{ν2(q−1), ν5(q−1)}=ν2(q−1) had been the case, the proof would be analogous by noting that ν5(q−1) ≥ν2(q−1). CLASS IV PRODUCTS Let q≡m≡11 (mod 20). We consider the product q·m, which belongs to the congruence class 1modulo 20. Following (2.2), we suppose that min{ν2(q·m−1), ν5(q·m−1)}=ν2(q·m−1),min{ν2(q+1), ν5(q−1)}= ν2(q+ 1), and min{ν2(m+ 1), ν5(m−1)}=ν2(m+ 1). From ν2((q+1)·m−m−1) = ν2((q+1)·m−(m+ 1)), we get ν2((q+1)·m−(m+ 1)) ≥ min {ν2((q+1)·m), ν2(m−1)}. Thus, ν2((q+ 1) ·m) = ν2(q+ 1) + ν2(m)so that ν2((q+1)·m)≥ν2(q+ 1). As a result, the 2-adic valuation of (q·m−1) is greater than or equal to the minimum between ν2(q+ 1) and ν2(m−1).